You effectively have a second-order equation, with a generalized function of the first derivative. You will need to be more specific about r(v) if you want to make any progress. If r(v) is linear, you're in luck - otherwise it may be quite tough to solve.

perhaps if you could give us some clue what the thrust term might be, we could explore it further.

The thrust equation is for the most part a type of step function: x newtons for 0.5s, y newtons for 2 sec, and z newtons for 4 secs. That's a bit hard to express in an equation unless we use peice-wise.

Which come to think of it may work.... if we solve the problem with T constant from t=0 to 0.5, then 0.5 to 2.5 and 6.5, each time carrying v0, perhaps we could solve it.

By the way - the equation is slightly doubtful - for a rocket you need to include the loss of mass due to fuel expenditure. Therefore m = m(t). You'll have to think about how this fits in - i.e., is the rate of fuel loss proportional to time?

I was using the assumption that the mass lost compared to the total mass of the vehicle was negligable in order to make calculations easier, but if I were to include it, it would be some function of the form

[tex]m(t)=m_{v} - km_{p0}t[/tex]

or the mass of the vehicle plus the initial mass of the propellant decreasing linearly with time.